Explicit Formulas

In a sequence to find any term without knowing the previous term, we use the explicit formula. It is the formula used to find the nth term based on its position. Let’s learn the explicit formula for different types of sequences:

To find the nth term of a sequence, we use the explicit formula, which can be an arithmetic, geometric, or harmonic sequence.  

nth term of an arithmetic sequence -  a_n = a + (n - 1)d, where
a is the first term

n is the position of the term in the sequence, 

and d is the common difference. Steps used for finding explicit formulas are -

Step 1: First find the first term and the common difference of the sequence

Step 2: Substitute the value of a, n, and d in the explicit formula, a_n = a + (n - 1)d

Step 3: Simplify the formula to find the nth term

To find the 7th term of the sequence 7, 14, 21, 28, … 

In the given sequence, a = 7 and d = 7(14 - 7 = 7)

The nth term of an arithmetic sequence is: a_n = a + (n - 1)d

So, the 7th term is: a7 = 7 + (7 - 1)7

= 7+ (6 × 7)

= 7 + 42 = 49

Therefore, the 7th term of the sequence is 49.

In an arithmetic sequence, the difference between any two consecutive terms is constant, and it is known as the common difference (d). To find the nth term of an arithmetic sequence, we use the explicit formula: an = a + (n - 1)d. 
Where, 

• a_n is the nth term
   
• a is the first term
   
• d is the common difference

For example, for the arithmetic sequence 2, 5, 8, 11, 14, …, finding the explicit formula
 
Here, a = 2

d = 3

The explicit formula of an arithmetic sequence is: an = a + (n - 1)d
Substituting the value of a and d:

a_n = 2 + (n - 1)3

= 2 + 3n - 3 

a_n = 3n - 1 

Find the 25th term of the sequence. 

a₂₅ = 3 × 25 - 1 

= 75 - 1

= 74

Therefore, the 25th term of the sequence is 74. 

In an arithmetic sequence, the difference between any two consecutive terms is constant, and it is known as the common difference (d). To find the nth term of an arithmetic sequence, we use the explicit formula: a_n = a + (n - 1)d. 

Where, 

• an is the nth term
   
• a is the first term
   
• d is the common difference

For example, for the arithmetic sequence 2, 5, 8, 11, 14, …, finding the explicit formula 

Here, a = 2

d = 3

The explicit formula of an arithmetic sequence is: an = a + (n - 1)d
Substituting the value of a and d:
a_n = 2 + (n - 1)3
= 2 + 3n - 3 
a_n = 3n - 1 

Find the 25th term of the sequence. 

a₂₅ = 3 × 25 - 1 

= 75 - 1

= 74

Therefore, the 25th term of the sequence is 74. 

The geometric sequence is any sequence where the ratio of any two consecutive terms is the same. The ratio is known as the common ratio (r). The general form of a geometric sequence can be represented as a, ar, ar², ar³, … ar^{n - 1}. For the geometric sequence, the explicit formula is an = ar^{n - 1}. 

Where, 

• a is the first term
   
• a_n is the nth term 
   
• r is the common ratio

For example, for the sequence: 1, 2, 4, 8, …, finding the explicit formula

Here, a = 1

r = 2

The explicit formula for geometric sequences is: a_n = ar^{n - 1}
Substituting the value of a and r:

a_n = 1 × 2^{n - 1}
= 2^{n - 1}

Finding the 5th term:

a_{5 =} 2^{(5 - 1)}

= 2⁴ = 16

So, the 5th term is 16 

The reciprocal of the terms in an arithmetic sequence is the harmonic sequence. For instance, the harmonic sequence of 2, 4, 6, 8, … is 1/2, 1/4, 1/6, 1/8, …. The general form of a harmonic sequence can be represented as 1/a, 1/(a +d), 1/(a + 2d), …, 1/(a + (n - 1)d). For a harmonic sequence, the explicit formula: a_n = 1/(a + (n - 1)d)

• Where a is the first term

• a_n is the nth term

• d is the common difference

For example, find the explicit formula for the harmonic sequence: 1/3, 1/6, 1/9, 1/12. 

We take the reciprocals of the terms: 3, 6, 9, 12, …, to find a and d. 
Here, a = 3

d = 3

a_n = 1/(a + (n - 1)d)

Substituting the value of a and d

a_n = 1/(3 + (n - 1)3)

= 1/(3 + 3n - 3)

= 1/3n

Finding the 5th term 

a_n = 1/3n

a₅ = 1/(3 × 5)

= 1/15

So, the 5th term of the sequence is 1/15

To calculate or predict a specific term in a sequence, we use the explicit formulas. Here are some real-life applications of the explicit formulas.

• In sports, we use the arithmetic sequence to design a workout routine to plan a steady increase in the repetitions. At any point in training to determine the number of repetitions, we use the explicit formula. 

• In finance, to calculate the savings, loans, and investments, where the amount forms a geometric sequence, we can use the explicit formula to predict the amount at a particular time. 

• While population growth is in a linear model, we can use the explicit formula to predict the growth at a specific time. 

• In computer algorithms, we can use explicit formulas to directly calculate the value in the sequence, which helps in improving efficiency.